Chapter 1: Introduction to Control Systems

What is a Control System?

A control system is a system that manages, commands, or regulates other systems to achieve a desired output.

Open-Loop vs Closed-Loop

graph LR
    subgraph Open-Loop
        I1[Input] --> C1[Controller] --> P1[Plant] --> O1[Output]
    end

graph LR
    subgraph Closed-Loop
        R[Reference] --> S((+))
        S --> C[Controller]
        C --> P[Plant]
        P --> Y[Output]
        Y --> M[Sensor]
        M -->|Feedback| S
    end

Key Differences

Aspect	Open-Loop	Closed-Loop
Feedback	No	Yes
Accuracy	Low	High
Disturbance rejection	Poor	Good
Stability	Usually stable	Can be unstable
Complexity	Simple	Complex

Transfer Functions

The transfer function relates output to input in the Laplace domain:

$G(s) = \frac{Y(s)}{U(s)}$

Example: First-Order System

$G(s) = \frac{K}{\tau s + 1}$

Where:

K = DC gain
τ = time constant

import numpy as np
import matplotlib.pyplot as plt
from scipy import signal

# First-order system: G(s) = 1 / (s + 1)
num = [1]
den = [1, 1]
system = signal.TransferFunction(num, den)

# Step response
t, y = signal.step(system)

plt.figure(figsize=(10, 6))
plt.plot(t, y, 'b-', linewidth=2)
plt.xlabel('Time (s)')
plt.ylabel('Response')
plt.title('First-Order System Step Response')
plt.grid(True)
plt.show()

Second-Order Systems

$G(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}$

Where:

ωn = natural frequency
ζ = damping ratio

Damping Cases

ζ Value	Response Type
ζ = 0	Undamped (oscillatory)
0 < ζ < 1	Underdamped
ζ = 1	Critically damped
ζ > 1	Overdamped

from scipy import signal
import numpy as np
import matplotlib.pyplot as plt

# Different damping ratios
omega_n = 1.0
zetas = [0.1, 0.5, 0.7, 1.0, 2.0]

plt.figure(figsize=(12, 6))

for zeta in zetas:
    num = [omega_n**2]
    den = [1, 2*zeta*omega_n, omega_n**2]
    system = signal.TransferFunction(num, den)
    t, y = signal.step(system, T=np.linspace(0, 15, 500))
    plt.plot(t, y, label=f'ζ = {zeta}')

plt.xlabel('Time (s)')
plt.ylabel('Response')
plt.title('Second-Order System - Effect of Damping Ratio')
plt.legend()
plt.grid(True)
plt.show()

Stability Analysis

A system is stable if its output remains bounded for any bounded input.

Poles and Stability

All poles must have negative real parts
Poles in the right half-plane → unstable
Poles on imaginary axis → marginally stable

import numpy as np

# Example: Check stability
num = [1]
den = [1, 3, 2]  # s^2 + 3s + 2 = (s+1)(s+2)

poles = np.roots(den)
print(f"Poles: {poles}")
print(f"Stable: {all(np.real(poles) < 0)}")